This is the analogue of “\(u\)-substitution” in higher dimensions. For a multivariable function \(f \colon \mathbf{R}^2 \to \mathbf{R}\) and the integral \(\iint_R f \,\mathrm{d}A,\) consider an invertible \(C^1\) transformation \(T \colon \mathbf{R}^2 \to \mathbf{R}^2\) for which \((x,y) = T(u,v)\) and let \(S\) be the preimage of \(R\) under \(T,\) so \({T(S)=R.}\) The Jacobian matrix \(\mathbf{J}_T\) of this transformation \(T\) is the matrix of partial derivatives of \(x\) and \(y\) with respect to \(u\) and \(v,\) and the Jacobian determinant \(\operatorname{J}_T\) (sometimes simply called the Jacobian) is the determinant of this matrix, which serves as an “area correction factor” when regarding the integral over \(S\) in the \(uv\)-plane that \(T\) transforms into \(\iint_R f \,\mathrm{d}A.\) \[ \mathbf{J}_T = \begin{pmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\[2pt] \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{pmatrix} \qquad {\color{maroon}\operatorname{J}_T} = \operatorname{det}\bigl(\mathbf{J}_T\bigr) %= \frac{ \partial(x,y) }{ \partial(u,v) } %= \operatorname{det}\begin{pmatrix} % \frac{\partial x}{\partial u} % & \frac{\partial x}{\partial v} % \\[2pt] \frac{\partial y}{\partial u} % & \frac{\partial y}{\partial v} %\end{pmatrix} = \frac{\partial x}{\partial u} \frac{\partial y}{\partial v} - \frac{\partial x}{\partial v} \frac{\partial y}{\partial u} \qquad \qquad \iint_R f \,\mathrm{d}A \;\;= \;\; \iint_S f\bigl(x(u,v), y(u,v)\bigr) \,{\color{maroon} \operatorname{J}_T} \,\mathrm{d}u\,\mathrm{d}v \] In three-dimensional space for a transformation \(T \colon \mathbf{R}^3 \to \mathbf{R}^3,\) the Jacobian matrix is a \(3\times 3\) matrix and the core idea is the same: \[ \mathbf{J}_T %= \frac{ \partial(x,y,z) }{ \partial(u,v,w) } = \begin{pmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\[2pt] \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\[2pt] \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \end{pmatrix} \qquad {\color{maroon}\operatorname{J}_T} = \operatorname{det}\bigl(\mathbf{J}_T\bigr) = \frac{\partial x}{\partial u} \frac{\partial y}{\partial v} \frac{\partial z}{\partial w} - \frac{\partial x}{\partial u} \frac{\partial y}{\partial w} \frac{\partial z}{\partial v} + \frac{\partial x}{\partial v} \frac{\partial y}{\partial w} \frac{\partial z}{\partial u} - \frac{\partial x}{\partial v} \frac{\partial y}{\partial u} \frac{\partial z}{\partial w} + \frac{\partial x}{\partial w} \frac{\partial y}{\partial u} \frac{\partial z}{\partial v} - \frac{\partial x}{\partial w} \frac{\partial y}{\partial v} \frac{\partial z}{\partial u} \] which corresponds to the integral equality \( \iiint_E f \,\mathrm{d}V = \iiint_S f\bigl(x(u,v,w), y(u,v,w), z(u,v,w)\bigr) \,{\color{maroon} {\color{maroon} \operatorname{J}_T}} \,\mathrm{d}u\,\mathrm{d}v\,\mathrm{d}w \,. \)
Jacobian conjecture (stated 1939, unsolved): For a transformation \(T\colon \mathbf{R}^n \to \mathbf{R}^n\) with polynomial components, if \(\operatorname{J}_T\) is non-zero and constant, then \(T\) has an inverse function \(T^{-1}\colon \mathbf{R}^n \to \mathbf{R}^n\) that is also defined by polynomial components.